An efficient algorithm for the single facility location problem with polyhedral norms and disk-shaped demand regions

André Berger,Alexander Grigoriev,Andrej Winokurow

doi:10.1007/s10589-017-9935-4

André Berger, Alexander Grigoriev + Show 1 more

Open Access

https://doi.org/10.1007/s10589-017-9935-4

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Abstract

The single facility location problem with demand regions seeks for a facility location minimizing the sum of the distances from n demand regions to the facility. The demand regions represent sales markets where the transportation costs are negligible. In this paper, we assume that all demand regions are disks of the same radius, and the distances are measured by a rectilinear norm, e.g. ell _1 or ell _infty . We develop an exact combinatorial algorithm running in time O(nlog ^c n) for some c dependent only on the space dimension. The algorithm is generalizable to the other polyhedral norms.

Highlights

In the conventional facility location problem each customer is associated with a single point in the space
In the basic problem considered in the present paper, we assume the transportation costs to a demand region are proportional to the shortest distance between any point in
In this paper we develop an exact algorithm for the single facility location problem with disk-shaped demand regions in Rd under rectilinear distance measures

Summary

Introduction

In the conventional facility location problem each customer is associated with a single point in the space. In the basic problem considered in the present paper, we assume the transportation costs to a demand region are proportional to the shortest distance between any point in. While Brimberg and Wesolowsky [3] have used the same cost function with the regions being polytopes, we consider a special case where the demand regions are disks (balls) of radius R ≥ 0 and the distances are measured by a rectilinear norm, e.g., 1 or ∞. Kumar et al [9] presented a randomized sampling algorithm, which for any ε > 0 with probability at least 1/2 finds a (1 + ε)-approximation to the geometric k-median problem in O(2(k/ε)O(1) dn) time, where d is the space dimension and n is the number of points. In this paper we develop an exact algorithm for the single facility location problem with disk-shaped demand regions in Rd under rectilinear distance measures.

Problem definition

A two-dimensional 1 case

Notation and definitions

Observations